Control and regulation problems in switching systems

Switching, or multimodal, dynamical systems are defined by a finite family of dynamics, namely the modes of the system, and a rule that governs the switching from one mode to another. Switching systems can model complex behaviours that originate when dynamical structures modify in response to varying operative conditions. Examples are given by multi-agent systems, where the set of active agents or that of communication links change during operation, as well as by mechatronic systems, whose sensor/actuator configuration odifies according to working conditions. Recently, switching systems with linear modes have been considered by many authors and different approaches to control and regulation problems for that class of systems have been proposed. Among them, the geometric approach, originally developed for linear systems, has proven to be particularly effective. Here, we discuss and investigate from the geometric point of view a number of control and regulation problems. Basic geometric concepts are introduced and illustrated, starting from the notions of invariance and controlled invariance. Then, the properties of internal and external stabilizability for controlled invariant subspaces are introduced and their role in constructing stabilizing compensators is highlighted. Solvability methods for disturbance decoupling, regulation and model matching problems, with additional stability requirements, can thus be stated and discussed.